bvode
bvode — boundary value problems for ODE
Calling sequence
[z]=bvode(points,ncomp,m,aleft,aright,zeta,ipar,ltol,tol,fixpnt,...
fsub1,dfsub1,gsub1,dgsub1,guess1)
Parameters
| z | The solution of the ode evaluated on the mesh given by points |
| points | an array which gives the points for which we want the solution |
| ncomp | number of differential equations (ncomp <= 20) |
| m | a vector of size ncomp. m(j) gives the order of the j-th differential equation |
| aleft | left end of interval |
| aright | right end of interval |
| zeta | zeta(j) gives j-th side condition point (boundary point). must have zeta(j) <= zeta(j+1)
all side condition points must be mesh points in all meshes used, see description of ipar(11) and fixpnt below.
|
| ipar | an integer array dimensioned at least 11. a list of the parameters in ipar and their meaning follows some parameters are renamed in bvode; these new names are given in parentheses. | ipar(1) | 0 if the problem is linear, 1 if the problem is nonlinear | | ipar(2) | = number of collocation points per subinterval (= k ) where max m(i) <= k <= 7 .
if ipar(2)=0 then bvode sets
k = max ( max m(i)+1, 5-max m(i) ) | | ipar(3) | = number of subintervals in the initial mesh ( = n ). if ipar(3) = 0 then bvode arbitrarily sets n = 5. | | ipar(4) | = number of solution and derivative tolerances. ( = ntol ) we require 0 < ntol <= mstar. | | ipar(5) | = dimension of fspace ( = ndimf ) a real work array. its size provides a constraint on nmax. choose ipar(5) according to the formula | | ipar(6) | = dimension of ispace ( = ndimi )an integer work array. its size
provides a constraint on nmax, the maximum number of
subintervals. choose ipar(6) according to the formula | | ipar(7) | output control ( = iprint ) | = -1 | for full diagnostic printout | | = 0 | for selected printout | | = 1 | for no printout |
| | ipar(8) | ( = iread ) | = 0 | causes bvode to generate a uniform initial mesh. | | = xx | Other values are not implemented yet in Scilab | = 1 | if the initial mesh is provided by the user. it is defined in fspace as follows: the mesh
will occupy fspace(1), ..., fspace(n+1). the user needs to supply only the interior mesh points fspace(j) = x(j), j = 2, ..., n. | | = 2 if the initial mesh is supplied by the user | as with ipar(8)=1, and in addition no adaptive mesh selection is to be done. |
|
| | ipar(9) | ( = iguess ) | = 0 | if no initial guess for the solution is provided. | | = 1 | if an initial guess is provided by the user in subroutine guess. | | = 2 | if an initial mesh and approximate solution coefficients are provided by the user in fspace. (the former and new mesh are the same). | | = 3 | if a former mesh and approximate solution coefficients are provided by the user in fspace, and the new mesh is to be taken twice as coarse; i.e.,every second point from the former mesh. | | = 4 | if in addition to a former initial mesh and approximate solution coefficients, a new mesh is provided in fspace as well. (see description of output for further details on iguess = 2, 3, and 4.) |
| | ipar(10) | | = 0 | if the problem is regular | | = 1 | if the first relax factor is =rstart, and the nonlinear iteration does not rely on past covergence (use for an extra sensitive nonlinear problem only). | | = 2 | if we are to return immediately upon (a) two successive nonconvergences, or (b) after obtaining error estimate for the first time. |
| | ipar(11) | = number of fixed points in the mesh other than aleft
and aright. ( = nfxpnt , the dimension of
fixpnt) the code requires that all side condition
points other than aleft and aright (see
description of zeta ) be included as fixed points in
fixpnt. |
|
| ltol | an array of dimension ipar(4). ltol(j) =
l specifies that the j-th tolerance in tol controls the
error in the l-th component of z(u). also require
that |
| tol | an array of dimension ipar(4). tol(j) is the error tolerance on the ltol(j) -th component of z(u). thus, the code attempts to satisfy for j=1,...,ntol on each subinterval
if v(x) is the approximate solution vector.
|
| fixpnt | an array of dimension ipar(11). it contains the points, other than aleft and aright, which are to be included in every mesh. |
| externals | The function fsub,dfsub,gsub,dgsub,guess are Scilab
externals i.e. functions (see syntax below) or the name of a Fortran
subroutine (character string) with specified calling sequence or a
list. An external as a character string refers to the name of a
Fortran subroutine. The Fortran coded function interface to bvode
are specified in the file fcol.f. | fsub | name of subroutine for evaluating
at a point x in (aleft,aright). it should have the heading [f]=fsub(x,z) where f is the vector containing the value of fi(x,z(u)) in the i-th component and
is defined as above under purpose .
| | dfsub | name of subroutine for evaluating the Jacobian of f(x,z(u)) at a point x. it should have the heading [df]=dfsub (x , z ) where z(u(x)) is defined as for fsub and the (ncomp) by (mstar) array df should be filled by the partial derivatives of f, viz, for a particular call one calculates | | gsub | name of subroutine for evaluating the i-th component of
at a point x = zeta(i) where
1<=i<=mstar.
it should have the heading[g]=gsub (i , z) where z(u) is as for fsub, and i and g=gi are as above. note that in contrast to f in fsub , here only one value per call is returned in g.
| | dgsub | name of subroutine for evaluating the i-th row of the Jacobian of
g(x,u(x)). it should have the heading [dg]=dgsub
(i , z ) where z(u) is as for fsub, i as for
gsub and the mstar-vector dg should be filled with the
partial derivatives of g, viz, for a particular call one calculates | | guess | name of subroutine to evaluate the initial approximation for
z(u(x)) and for dmval(u(x))= vector of the
mj-th derivatives of u(x). it should have the heading
[z,dmval]= guess (x ) note that this subroutine is used
only if ipar(9) = 1, and then all mstar
components of z and ncomp components of dmval should be
specified for any x, aleft <= x <= aright . |
|
Description
this package solves a multi-point boundary value
problem for a mixed order system of ode-s given by
the boundary points satisfy
the orders mi of the differential equations satisfy
1<=m(i)<=4.
Examples
deff('df=dfsub(x,z)','df=[0,0,-6/x**2,-6/x]')
deff('f=fsub(x,z)','f=(1 -6*x**2*z(4)-6*x*z(3))/x**3')
deff('g=gsub(i,z)','g=[z(1),z(3),z(1),z(3)];g=g(i)')
deff('dg=dgsub(i,z)',['dg=[1,0,0,0;0,0,1,0;1,0,0,0;0,0,1,0]';
'dg=dg(i,:)'])
deff('[z,mpar]=guess(x)','z=0;mpar=0')// unused here
deff('u=trusol(x)',[ //for testing purposes
'u=0*ones(4,1)';
'u(1) = 0.25*(10*log(2)-3)*(1-x) + 0.5 *( 1/x + (3+x)*log(x) - x)'
'u(2) = -0.25*(10*log(2)-3) + 0.5 *(-1/x^2 + (3+x)/x + log(x) - 1)'
'u(3) = 0.5*( 2/x^3 + 1/x - 3/x^2)'
'u(4) = 0.5*(-6/x^4 - 1/x/x + 6/x^3)'])
fixpnt=0;m=4;
ncomp=1;aleft=1;aright=2;
zeta=[1,1,2,2];
ipar=zeros(1,11);
ipar(3)=1;ipar(4)=2;ipar(5)=2000;ipar(6)=200;ipar(7)=1;
ltol=[1,3];tol=[1.e-11,1.e-11];
res=aleft:0.1:aright;
z=bvode(res,ncomp,m,aleft,aright,zeta,ipar,ltol,tol,fixpnt,...
fsub,dfsub,gsub,dgsub,guess)
z1=[];for x=res,z1=[z1,trusol(x)]; end;
z-z1
See also
fort, link, external, ode, dasslAuthor
u. ascher, department of computer science, university of british; columbia, vancouver, b. c., canada v6t 1w5; g. bader, institut f. angewandte mathematik university of heidelberg; im neuenheimer feld 294d-6900 heidelberg 1 ; ; Fortran subroutine colnew.f
